Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Possible Duplicate:
Universal Chord Theorem

Let $f:[0,1] \to R$ be a real valued continuous function satisfying $f(0)=f(1)$. Then using intermediate value theorem we know for every $n \in N$ there exist two point $a,b \in [0,1]$ at a distance $1/n$ satisfying $f(a)=f(b)$.

Now my question is, for every $r\in [0,1]$ is it possible to find two points $a,b\in [0,1]$ at a distance $r$, satisfying $f(a)=f(b)$ provided $f:[0,1] \to R$ be a real valued continuous function satisfying $f(0)=f(1)$.

as there is a counterexample for $r>1/2$, please consider the case when $r<1/2$.

share|improve this question

marked as duplicate by Martin Sleziak, William, Noah Snyder, Hagen von Eitzen, Henry T. Horton Oct 9 '12 at 21:50

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

@martin sorry martin but i was not aware of the universal chord theorem. And thanks for your references. –  timon Sep 20 '12 at 17:34
add comment

1 Answer

up vote 1 down vote accepted

Hint: Consider $r=\frac23$ and $$f(x)=\begin{cases}x&\mathrm{if\ }x\le \frac13\\ 1-2x &\mathrm{if\ } \frac13<x<\frac23\\ x-1 &\mathrm{if\ }x\ge\frac23 \end{cases}$$

share|improve this answer
thank you for your answer.I have one doubt. For $r< 1/2$ will we be able to produce counterexample. –  timon Sep 20 '12 at 16:54
add comment

Not the answer you're looking for? Browse other questions tagged or ask your own question.